Portfolios: Expected Return and Risk

Kerry Back

BUSI 721, Fall 2022
JGSB, Rice University

Review of Diversification

Portfolio mean is weighted average of asset means

Portfolio std dev < weighted average of asset std devs

3A

Calculating Portfolio Variance

\(\sum_{i=1}^n w_{i}^2var(r_i)+2\sum_{i=1}^n\sum_{j=i+1}^n w_{i}w_{j}cov(r_{i},r_{j})\)

Let C = covariance matrix, w = vector of weights.

Portfolio variance is \(w^{⊤}Cw\).

Can calculate covariance matrix as SRS where S is diagonal matrix containing std devs and R is correlation matrix.

import numpy as np

# standard deviations
sds = np.array([0.20, 0.12, 0.15])
S = np.diag(sds)

# correlations
r12 = 0.4
r13 = 0.2
r23 = 0.3

R = np.identity(3)
R[0, 1] = R[1, 0] = r12
R[0, 2] = R[2, 0] = r13
R[1, 2] = R[2, 1] = r23

# covariance matrix
C = S @ R @ S

# portfolio
w = np.array(0.25, 0.25, 0.5)

# portfolio std dev
np.sqrt(w @ C @ w)

3B

Risk-Free Saving

Suppose asset B is risk free. Set \(r_{f}=r_{B}\).

Then, \(\small var(r_{p})=w_{A}^2var(r_{A})\) and \(\small stdev(r_{p})=w_{A}stdev(r_{A})\)

\[\small \bar{r}_p=w_{A}\bar{r}_{A}+(1-w_{A})r_{f}=r_{f}+w_{A}(\bar{r}_{A}-r_{f})\]

Portfolio risk premium is: \[\small \bar{r}_p-r_{f}=w_{A}(\bar{r}_{A}-r_{f})\]

So both risk and risk premium scale with allocation to risky asset.

Sharpe Ratio

The Sharpe ratio is the ratio of risk premium to risk:

\[\frac{\bar{r}-r_{f}}{\sigma}\]

When combining a risk-free asset with a risky asset, the Sharpe ratio is invariant to the risky asset allocation.

The Sharpe ratio of the portfolio is the Sharpe ratio of the risky asset.

Buying on Margin (Leverage)

Continuing with asset B = risk-free, we could take \(w_{A}>1\), meaning that \(w_{A}-1\) is borrowed.

Same formulas apply:

risk premium is \(w_{A}\) times A’s risk premium,
st dev is \(w_{A}\) times A’s std dev.

Borrowing rate will be higher than saving rate.

3F

Borrowing or Saving with More Assets

\(n\) risky assets. Allocation to risk-free asset is \(1-\sum_{i=1}^n w_{i}\).

Portfolio return is:

\[r_{p}=\sum_{i=1}^n w_{i}r_{i}+(1- \sum_{i=1}^n w_{i})r_{f}=r_{f}+\sum_{i=1}^n w_{i}(r_{i}-r_{f})\]

So, risk premium is weighted average of asset risk premia.